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disha: searching on "Andrew Yao" - A Turing Award winner - will yield more results. trapdoor and one-way functions seem to be synonymous. I remember seeing a definition that trap door functions in addition to being one-way functions need to have a way to get original message out with some extra information. This is for law enforcement purposes.

I don't thnk bhaskara had a solution to perfect encryption,I.e. A true one way function. Existence or not of a one way function is unsolved AFAIK.

Is it believed that there exists no one way functions? Then bhaskara couldn't have gotten one.

The problem was asked as a challenge to JohneeG as an experiment on my part. He started a thread "Sciences in Ancient Bhaarath: Truth vs Myth". The similar topics have appeared in Brf (and other blogs) many many times and apart from a few useful posts, mainly one sees two kind of people arguing ..

A) Those who think that there was ZERO contribution from India. (All science developed in west ... apart from cast system there was nothing else in India-- etc)

B ) Every thing was invented/discovered in India.. Vedic science knew/discovered everything from an Atom Bomb to Age of the Universe ... not to mention Hanuman knowing exact distance of the sun -- Hanuman Chalisa as the proof -- so that he can "jump" and try to eat the sun (a 5000 degree surface - and 1000000 times the volume of Earth) as a fruit. ....(This was a JohneeG's post. When I mentioned that the distance is NOT measured in "years" -- his response was that he made a typo in saying "yuga=12000 years" because yuga is just a number 12000 and not 12000 years and he ridiculed me. ). Of course, he did not say how Hanuman "eat" the sun when math will tell you that, up that close, both sun's size and temperature will be much different than its apparent size from earth..

Anyway both these types, in my view are so wrong, that it is absurd. And one must have a shocking lack of even the most basic math principles to advocate such points. I wanted to do an experiment on both these types, by asking a simple math problem. To see if these "experts" even with the help of internet or their own resources can even understand basic math... I think the second kind (those who come out with gems like Tulsidas was astronomer because he predicted Hanuman eating sun) are even more absurd and dangerous if they are allowed to brainwash the young.

Interestingly the problem posed by me DOES deal with Bhaskara (and Brahamgupta) method(s). If they really studied Bhaskara they will UNDERSTAND the Maya in that Baskara's wheel...

***

Coming back to the problem (to find integer solution for y^2=x^2+7)..- As said before, Bhaskara's/ Brahmgupta math may be sufficient to find a solution.- Pay attention to "maya" and the word "said" .. that a number was found"...- So a solution would be okay to show that it is indeed a "maya" - (as Disha suspects).. but one has to PROVE that in that case no solution exist. - The problem is kind of problems which were favorite of Ramanujan ... along with much famous problem from his notebook -- to find a solution to "(2^n-7 as a perfect square" ... see: here - brf math dhaga is a good resource - Actually this class of equations is studied by another famous mathematician who was with Ramnujan in Cambridge, and this kind of equation is named after him. (This mathematician is actually famous for proving some of Ramanujan's results).. (I am not naming the Mathematician's here.. just to make googling a little more difficult ) ..

There is a small history behind this problem .. It was "short listed" for a international math contest about 10 years ago but did not appear in actual contest because some thought that it may be too easy for that level.

***I have nothing against those who really want to study Indian Math history, try to educate others if they found some good points, and ask valid question or debate valid points if some one disagrees or raises a point. So I am not disrespecting anyone including JohneeG.. and I hope people do not take pointing out absurd points as disrespect.

disha wrote:However it reminds me of the equation y^2 = x^3 + ax + b which is an elliptic curve and an excellent trapdoor function.

Nice !!!

You may have already realized this but the same kind of technique can be applied here too.

Another thing, which I am sure you know that, except for trivial case, there are only finite number of solutions. (for general y^2=x^3+k) ... which makes it easy to find a solution specially using a computer..

Thanks Disha .. let me take this opportunity to introduce another Math genius from India S S Pillai (Subbayya Sivasankaranarayana Pillai)... some compared him as next to Ramanujan -- He was 10-20 years younger than Ramanujan. (Vishvakji - Please, if you have not done it, do read about him and write something here in this dhaga - as I am sure you like to do)

His famous conjuncture is interesting (it is deals with more than cubic part)

In layman's language it says.. There are only FINITE values which satisfies..x^m=y^n+k When m,n>2...

So y^2=x^7+7 ONLY has finite number of solutions.( actually None in this particular case) (See ; Wiki entry

***

BTW, one of the famous problem is Y^2+2=x^3 which has only ONE solution (25+2=27).

AmberG: What you described is the scientific method. Indian science was scientific for its day. What people are forgetting or failing to see through the fog of religiosity - which has its place, of course, but not in hard sciences - is that there is always progress as new observations are made which are expressed in mathematical language so that predictions can be made as well as the machinery can be constructed to better lives.

Is that person you are talking about at Oxford Roger who did work on partition functions? He proved several identities which Ramanujan re-derived independently?

I think anybody who likes to understand Ramanujan's life should - at the minimum - read Kanigel's book. It has a very good description of intellectual life at Cambridge where Mahalanobis was studying as wll as Russel, Whitehead (Russel's adviser), GH Hardy, Littlewood including several other luminaries were dons.

I also posted about Narayana bhatta in relation to Combinatorics, pingaLa's chandas and its relation to deBruijn graphs. Interestingly deBruijn graphs can be used to set up experiments for functional MRI studies. They can be used to study a sequence of stimuli-and response using minimize number of functional MRI images. But anybody who claims that pingaLa (200 or before BCE - 100 AD) had known about fMRI will be laughed out from any group of rationalists irrespective of their religious beliefs or lack thereof.

Coming to the important part

I suppose there is a proof that there is no integer solution for

y^7 + 7 = x^2

Is it based on the fact that gdc(2,7) = 1?

OK. So according to Pillai's there are finite number of solutions. Do m and n hav to be co-primes, i.e. gcd(m,n) = 1 is required?

well my brute force - i.e. a simple program for the above did not find any - it exited due to numbers being too large as i did not really write it too robustly; so was going to say I doubt this has any solutoin .

thanks for the general exercise - came to know a few things. ---x--

At the same time I think you are only looking at the extermes. Most are in-between; believe that there was strong contribution to scientific and mathematical knowledge from Indians of the past. However, most of the text books or popular science books, at a maximum make a passing reference to the Indian contribution and revert to the standard Greek - Roman - European framework; and also posit that scientific temper is a European gift to the world. Thus there is a need to both understand what was contributed and also to recognize how the world came to a common modern state of knowlege - not a greek, indian, chinese, european knowlege.

---x----

People having extreme views is not limited to in India. Ive met highly accomplished engineers - creme de la creme - in multi-nationals who believe in intelligent design.

Another one is Kak who might go overboard sometimes ...Kak has the right attitude - though he is extending himself here: http://arxiv.org/pdf/physics/9804020 to state speed of light being known in Vijayanagara times. It would be better if he coudl find some description of how the value on 2022 yojanas in half a nimisha was determined (apparently, that comes to speed of light as per the paper).

Amber G. wrote:BTW, one of the famous problem is Y^2+2=x^3 which has only ONE solution (25+2=27).

I'm aware of that one because of Simon Singh's book on Fermat's Last Theorem. If I remember correctly, it was stated in a slightly different format by Fermat himself. Something along the lines of 26 is a number that lies in between a square (25) and a cube (27). Are there any other numbers that do this or is 26 the only one? Effectively he was stating that he wanted all integer solutions for Y^2 + 1 = X^3 - 1 (later on in the book, the author talks about elliptical equations and gives Y^2+2=x^3 being an example and on the second or third reading of the book, it occurred to me that this was the same equation that Fermat had proposed earlier, just with the terms slightly rearranged)

Fermat claimed that he had a proof that 26 was the only number that satisfied this condition and challenged other mathematicians to prove him right or wrong. Apparently he pissed off a bunch of mathematicians with this particular challenge (Descartes called him "the braggart" and John Wallis called him "the damned Frenchman").

One more interesting thing I learned was that while Fermat is most famous for his infamous quote:

Fermat wrote:I have a truly marvellous demonstration of this proposition, which this margin is too narrow to contain…

It wasn't the only time he wrote things like that. He did it all the time. Some of his other writings also went along the same vein.

Fermat wrote:I can prove this, but I have to feed the cat

I can solve this equation, but I have to wash my hair

Apparently, he liked throwing challenges to professional mathematicians all the time. Being a true Frenchman, he most particularly liked to torment English mathematicians , Wallis and Digby hated him for that.

ArmenT - Thanks for the interesting Fermat's story. I have got to read Simon Singh's book..

I think the problem (x^2+2=y^3) is much older but I did know this one was Fermat's "other case" where he claimed a proof which he did not have!

If any one is curious, here is the actual image of the page on the book.Image of the page

Fermat claimed that the only solution is the obvious one and conjectured (and hints that he knew how to prove it, but without explicitly saying so) that this can be proved by descent. I am fairly sure that Fermat, if he really believed to have a proof (in my opinion he did not) was mistaken.

I have tried to prove this by using Fermat's techniques but was not able to do it - come very close but no success - and I have not seen any other proof based on his technique (infinite decent type) technique alone.

Euler provided a proof of this, but it may have a flaw (or an omitted step)..

Another very good readable part I found is from Cox here: (It deals with the above problem - see page 2 - in and its relationship with Ferma'ts last theorem etc.. any way a very good read.http://math.stanford.edu/~lekheng/flt/cox.pdf

***

The proof of this is quite elegant and fun (I may put it in separate post - as it needs very little advance math and can be followed ).

Using the same technique one can prove that there is NO solution forx^2=y^3+7..

which is not that far away from x^2=y^7+7 ArmenT: Looks like you found a cool solution, as you posted in the other thread.. would you like to share it here..

Hilbert who tried very hard to get her a faculty position, could not overcome the resistance of the humanities professors, who simply could not stomach the idea of a female teacher. At one meeting of the faculty senate, frustrated again in his attempts to get Noether a job, he famously remarked, “I do not see that the sex of a candidate is an argument against her admission as Privatdozent. After all, we are a university, not a bathing establishment.”

Einstein wrote in NY times at the time of her death .. (Noticing that many of her students got fame but she remained quite unknown..

Emmy Noether, who, in spite of the efforts of the great Göttingen mathematician, Hilbert, never reached the academic standing due her in her own country, none the less surrounded herself with a group of students and investigators at Göttingen, who have already become distinguished as teachers and investigators.

But even if the public wasn’t aware of Noether’s greatness and passing, the mathematics community certainly was. Fellow Göttingen mathematics great Hermann Weyl delivered a moving eulogy at Noether’s funeral.

You did not believe in evil, indeed it never occurred to you that it could play a role in the affairs of man. [...] in a sea of hate and violence, of fear and desperation and dejection — you went your own way, pondering the challenges of mathematics [...] When you were not allowed to use the institute’s lecture halls you gathered your students in your own home. Even those in their brown shirts were welcome [...] Many of us believed that an enmity had been unleashed in which there could be no pardon; but you remained untouched by it all

US team (since 1994) "beat" China and came in first...!(They got 5 Golds and 1 Silver)

(Time passes very quickly - it only seemed a few years since the head coach Po-Shen Loh (he, his brother and sister all were past IMO medalist) was a contestant in a Math contest which I attended as a coach..)

We seriously need to improve our standards in euclidean geometry and mathematical reasoning... we need to introduces number theory at the school level...IIT coaching has ensured that we continue to fair poorly...AmberG can u post link to this years IMO questions ?

historically we have only done well in Physics (usually top 5) ...somtimes in astronomy and chemistry..Biol has been mixed...We have usually failed the worst in math...Most of it is because of our emphasis on rote memorization and mechanical solving of problems rather than creativity and reasoning...other reason is the curriculum ... Our pre-engineering/Pre-medical curriculum is calculus heavy ....we don't teach kids any number theory or euclidean geomentry (beyond simple problems from class 6 to 10) or mathematical logic...other thing is that we don't quite have decent state level competitions...In my days , I bought a problems primer...I could not solve more than 15% questions... I had not learned any number theory... my math tutor , who has taught JEE for Bansal etc could barely manage a quarter problems himself...and many 12-13 year olds in china and US can solve a good number of them...problem is we are not used to that kind of reasoning in school level...I later realized that many of the problems were quite easy...I was just not thinking the way I should have...

A few years back I had quick look at Russian 10th grade topology three month summer camp. The standards are very high. Atiyah-Ramanujam-Singer Index theorem, Ricci flows are dealt with over a three month long summer course.

IMHO, solving IMO problems or other "set for competition" problems are not as important as being able to push the envelope. But the solving IMO question bank problems certainly helps in honing one's skill in formal proofs and some amount of memoization (not memorization - but more in the sense of memoization in Dynamic Programming and Algorithms akin to the ability to keep a lot of terms/definitions/lemmas in one's short/long term memory).

To bridge the lacunae I think JEE should introduce IMO type questions. I think we will start seeing results in a few years. The IIT coaching system will be able come up with the creativity required to industrially solve such things.

^^ IMHO that is not the whole point..The objective should not be doing well in the olympiads in itself...China , North Korea and Iran too usually do well...But in their cases it is not reflective of the scientific prowess..

If we do like they do , ie select talented kids specially for the olympiads , and train them for 4 years , we can surely win all golds..But that does no good for the country..

In India's case the olympiad performance is reflective of the standard of math/science education...

In 7-8 grade one might remember having a chapter called goemetrical constructions.. most kids mechanically learnt have to construct bisectors /trisectors of lines , angles etc...most will not know , why a series of steps would bisect an angle...(it is kind of impossible to trisect an angle..except some angles like pi/2 and pi)

that refelcts our education system..

to compound the problem , there is a misconception prevalent in India ...That our schooling system is the toughest in the world and our kids learn more than western kids etc...surely we cram a lot more into the head and our kids spend more time on the study table than they should ...but that does not mean they are truly learning stuff...Our system lacks depth and focuses on trivia...

There should not be a one size fits all model for education...I am sure not every Russian 10th grader is made to learn the atiyah-singer index theorem etc...

Vriksh wrote:To bridge the lacunae I think JEE should introduce IMO type questions. I think we will start seeing results in a few years. The IIT coaching system will be able come up with the creativity required to industrially solve such things.

Vriskh (and Atri: Where art thou?): on another note, what is the progress of your bio-startup? there was thread on BRF which has been in suspended animation after a couple of months of frantic activity.

Off topic. As far as our bio-startup is concerned: we are up and running and have multiple projects India wide specifically projects in Ahmedabad, Delhi, Mumbai, Bangalore and some upcoming in Assam and Jammu. We have demonstrated that we can robustly and completely solve the problem of water, waste water and sanitation in India and hopefully abroad (I was eyeing So-Cal, ME and other territories). Will write more about opportunities and challenges we face in the Entrepreneurship thread.

For Atiyah-Singer Index Theorem is referred to as Atiyah-Patodi-Singer Index Theorem which I termed incorrectly Atiyah-Ramanuam-Singer index Theorem. Prof. Vijay Kumar Patodi of TIFR (who passed away at an young age of 31 just after one year of becoming a full professor) joint work with Atiyah and Singer at IAS led to the proof of the theorem.

interestingly, USSR -> 14, Russia -> 2.. then to which broken USSR country did the dozen medal went?

In my humble opinion some of the posts above may be missing a few points.. a few comments..The "ranking" or who (which country) "won" IMO (Math Olympiad) is silly even though most dorky media reports it as such. Of course, officially there is NO such ranking. Apart from media's shallow reporting it caries really no meaning. ..

It is a individual contest where kids get to solve/enjoy some of the most challenging problems.. some get gold (or silver or bronze) medals but even those who got even 1 point has noting to be "ashamed" about. After all these 6 kids have been selected from literally millions of children and are top in their field. Trust me, even the person with lowest "score" is (as far as her future -- can get int to the best university in the world easily --== to say someone ranked 6 in all india JEE getting into IIT).

Honestly it is silly to despair that India is ranked XXX... What to me is more serious is VERY few schools in India (all schools can take part but most do not know about it) do not participate in RMO/InMO (similar to AMC/USMO)..What I will like to see, like US where 100s of thousands (or may be million or so) take part in those qualifying tests.. .. so that, they can discover, if they are interested in Math..and instead of getting lost is very poor standard of Math in regular high school/college are exposed to fun/creative part of math which will make them better Engineers/doctors/citizens etc ..by an order of magnitude...

For me, I have been fortunate to have a father who gave me the love for learning and a world class guru to teach/challenge me math fun problems when I was young..so I enjoy more than anything else giving back by teaching young kids..(This Friday I had a pleasure of listening to Sal Khan of Khan Academy and was really inspired to hear how rewarding this could be)

Anyway .. let me just end with three you tube videos.. of IMO 2015... Enjoy...

AmberG ji: I have two books on IMO kids - one was where the author actually lived with the US IMO team during the time they were preparing for the big one. I will post the links.

Secondly, there is an MAA (or is it AMS) site where a majority of Chavenet Prize (it is given to outstanding expository writing by a Mathematics Professor) works are linked in the section listing the winners of that prize. I read parts here and there - the one which caught my eye was the paper by Ravi Vakil titled "The mathematics of doodling", very ingenious way to think about doodling.

vayu tuvan wrote:AmberG ji: I have two books on IMO kids - one was where the author actually lived with the US IMO team during the time they were preparing for the big one. I will post the links.

As you probably know I have been involved with "IMO kids" (and similar other groups) so this kind of interaction is one of the most rewarding experience..lot of anecdotes .. a few "kids" are now very accomplished human beings .. but still keep in touch..and frankly one of the best learning experience for the teachers..

The network/friendship formed between kids (and coaches) at the IMO type events last a life time.. many kids, for the first time (regular school, in most cases, really do not cater to them and if anything they are branded "not normal" and they do not fit, in any case they do not have knowledgable teachers/mentors to be helpful) meet others who are also interested in math..

***From above, the problems like the above one are the one which hooked me to math.. as I said in that post:

Manjul's Q&A brought back good memories... When I was quite young.. I attended a public lecture titled "Beauty of Mathematics" which was given at the local university but I went even though I was much younger. Prof. Mehta, introduced these numbers in very beautiful way and I still remember the excitement of learning new things...I was hooked.

Note 1: (if one used f(n)=f(n-1)+f(n-2)(which uses a different characteristic eq) the numbers are famous Hemchandra Numbers (Fibionacci)***For those who are interested in number theory (specially computer/numerical calculation algorithms):

==> The fastest way to compute (which computers use) is repeated squaring, not unlike method shown by Chilarai but this reduces the number of operation to about the order of log_2(n) instead of (n)..(That is if one wants to find, say f(10000000000) the above method needs about 10000000000 steps, repeated squaring reduces this to about 30 steps only ..

(This kind of methods are used in, among other uses, primality testing and other practical uses)

Amber G. wrote:==> The fastest way to compute (which computers use) is repeated squaring, not unlike method shown by Chilarai but this reduces the number of operation to about the order of log_2(n) instead of (n)..(That is if one wants to find, say f(10000000000) the above method needs about 10000000000 steps, repeated squaring reduces this to about 30 steps only ..

(This kind of methods are used in, among other uses, primality testing and other practical uses)

As it happens I am currently doing security testing on a chip which has a practical use of similar algo. It is to perform fast exponentiation ( for RSA ) on a smart card with limited memory and processing power.

to calculate x^10 , you can either perform 9 multiplications or use the square and multiply algo to do it in 1010

(((x^2)^2)*x)^2 which needs 4 multiplication.

It is in fact possible to do better than log_2(n) using addition chain exponentiation.( https://en.wikipedia.org/wiki/Addition_ ... nentiation ) but with memory in smart card being limited , does not allow storing of all intermediate values .Plus in bank card the implementation needs to be resistant to side channel analysis ( this is my line of work) , so if the power profile of a square operation is different than multiply operation , it can be possible to reveal the exponent ( RSA private key) by measuring power consumption during a payment transaction even though RSA itself is mathematically secure, so there are some nifty techniques involving splitting the exponent into parts and then masking them with random numbers in ways that the final results stays the same but the intermediate values are not related to the exponent.

^^^ Yes, this kind of application (which is easy to understand even by a high school student - but IMO 99% of even the college math majors are not familiar with it)produces beautiful math...

For example, the number 170141183460469231731687303715884105727 (this is nothing but 2^127-1)was proved to be prime by Lucas (in 1876 !!! and held the record till the age of computers (even Ramanujan or other great mathematicians did not improve on this) after 1950) using just hand. Thus instead of doing >> 1000's of trillion operations - (which is still a BIG deal even with FASTEST computers -- years/centuries of a super computer).. the person did it by hand (close to 127 operations!)

(For perspective greats like Euler were able to prove primality of a 10 digit number only (2^31-1)And when computers came, a high school student - who learned this method) made history by discovering larger prime number in first days of his of computer program writing - just after he learned the technique from his math-teacher !!)

****

Personally here is an anecdote: I was asked by a Jain Guru to verify if 11111111111 is prime or not.. (this was in 1960's and they knew I had access to computers in IIT/K). Old Jain mathematicians knew 1111111 was not a prime (divisible by 239) but no one knew if 1 (11 times) is prime or not.

Computer, of course, spilled out the result in seconds .. (11111111111 is not a prime - can you prove this using a simple program or a calculator?)

But later, I performed a simple test, and found out (without a computer or even a calculator) that the number is definitely not a prime. (And they could have performed this even 1000's of years ago, only if hey knew these kind of technique)

***

For those who want to learn - (NO higher math is needed just high-school math..)

You find 2^1111111110 and divide this by 11111111111 worry only about reminder. if the remainder is NOT 1, the number is not prime.

(The technique is: for any number n , find 2^(n-1) mod n, if the answer is NOT 1 then the number n is not prime)Try this out for n = 3,4,5,6, 7 etc to see what I mean)

(The point is to check for 11111111111 , you only have to do about 30 (about log n) operations which one can do by hand and NOT millions where one do need a computer)..

Hope some find it interesting...(BTW basic principles of RSA type encryption is this kind of math)

x-post - About 2 weeks ago, I have posted a small challenge in brf_math dhaga (<link here>to test the resourceful ness of brf community. I also believe those who like logical thinking will find the problem interesting.

It is a high-school level problem, easy to define and uses only the most basic math concepts.Lot of interesting discussion has already been there in that dhaga.

Here is what I said, wrt to the problem: Please discuss the problem in GDF dhaga, (so that the discussion is little free and among brf community only)

Amber G. wrote:I always liked the elegance and simplicity of Math. The kind where one does not use fancy terms or mumbo-jumbo yet logic is beautiful and could be understood with minimum amount of background. Though, of course, as a physicist I have to learn math, take college courses etc, even learn fancy terms , yet I still find elegant problems still fascinating. Problems like FLT, or Four Color problem -- easy to state with elementary school background.

So in that spirit, let me put a challenge in a form of a problem here. Whether your math background is simple school (basic arithmetic operations and natural numbers) type or advanced --you consider yourself a hot-shot and experienced in many techniques -- I hope all find this problem easy to understand, simple (easy)looking, yet challenging to all.

Feel free to try it, spend time with it, ask others (specially math professors, to see how good they really are /smile/) and enjoy. I will wait for at least a month to see how good/resourceful the math-community in BRF is.<problem quoted only in GDF dhaga>

(Feel free to discuss inside GDF but do not post it or discussion posted results here outside brf as I may use this or similar problem )